In this paper, the distributed cooperative control problems of discrete-time second-order networks over finite fields are investigated, under which a new perspective is provided for discrete-time second-order networks of agents with limited capacities of memory and communication. Firstly, the discrete-time second-order multi-agent model with a given protocol is established over the finite field. Then the leaderless finite-field synchronization network is studied, some necessary and sufficient conditions along with the design principles of the synchronous protocol are given to guarantee the network synchronization. In addition, the finite-field consensus problems including leaderless and leader-following cases are studied in sequence, both of which prove to be unachievable under the distributed control protocol. Finally, an application on camera networks and numerical examples are provided to support the obtained resultsNote to Practitioners—The multi-agent system is a collection of multiple agents with computing and mobility capabilities, and its distributed cooperative control problem aims to achieve the global objective by neighbor-based protocols, mainly including consensus, synchronization, formation control, etc. Because of its strong fault tolerance, robustness and scalability, it is widely used in military, industrial and agricultural production, medicine, transportation and so on. These three factors that mainly affect the group behavior of agents are the dynamic evolution, the network topology (that is, the connection between agents), and the communication mechanism. However, in practical applications, due to limited computation, memory, and communication capabilities, the state of the network is constrainted, which directly affects the dynamic evolution of the system. In this paper, a new kind of second-order synchronization dynamics, which can be used in certain applications such as multi-agent systems with periodic boundary conditions on position and velocity, is proposed. The states and communication weights of agents can only attain values from a finite alphabet equipped with a prime number of integers, called finite fields. Such networks have a finite convergence time, and are therefore friendly to multi-agent systems with limited capacities.

Observability ensures that any two distinct initial states can be uniquely determined by their outputs, so the stream ciphers can avoid unobservable nonlinear feedback shift registers (NFSRs) to prevent the occurrence of equivalent keys. This paper discusses the observability of Galois NFSRs over finite fields. Galois NFSRs are treated as logical networks using the semi-tensor product. The vector form of the state transition matrix is introduced, by which a necessary and sufficient condition is proposed, as well as an algorithm for determining the observability of general Galois NFSRs. Moreover, a new observability matrix is defined, which can derive a matrix method with lower computation complexity. Furthermore, the observability of two special types of Galois NFSRs, a full-length Galois NFSR and a nonsingular Galois NFSR, is investigated. Two methods are proposed to determine the observability of these two special types of NFSRs, and some numerical examples are provided to support these results.